Problem 25 - Brief Applied Calculus

Question

Find \(f^{\prime}(x)\) by using the definition of the derivative. [Hint: See Example 4.] $$ f(x)=x^{2}-3 x+5 $$

Step-by-Step Solution

Answer

The derivative is \( f^{\prime}(x) = 2x - 3 \).

Step 1: Recall the Definition of Derivative

The derivative of a function \( f(x) \) at a point \( x \) is given by the limit formula: \[ f^{\prime}(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}. \] This definition will be used to find the derivative for the function \( f(x) = x^2 - 3x + 5 \).

Step 2: Substitute into the Definition

Substitute \( x+h \) into the function to get \( f(x+h) = (x+h)^2 - 3(x+h) + 5 \). Expand this and simplify to find \( f(x+h) - f(x) \).

Step 3: Simplify \( f(x+h) \)

Compute \( f(x+h) = (x+h)^2 - 3(x+h) + 5 \). Expanding gives: \[ f(x+h) = x^2 + 2xh + h^2 - 3x - 3h + 5. \]

Step 4: Compute \( f(x+h) - f(x) \)

Subtract \( f(x) = x^2 - 3x + 5 \) from \( f(x+h) \) to obtain: \[ f(x+h) - f(x) = (x^2 + 2xh + h^2 - 3x - 3h + 5) - (x^2 - 3x + 5). \] Simplifying yields \( f(x+h) - f(x) = 2xh + h^2 - 3h \).

Step 5: Divide by \( h \) and Simplify

According to the limit definition, divide the result by \( h \):\[ \frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 - 3h}{h} = 2x + h - 3. \]

Step 6: Take the Limit as \( h \to 0 \)

Finally, take the limit of the expression as \( h \to 0 \): \[ \lim_{{h \to 0}} (2x + h - 3) = 2x - 3. \] This gives us \( f^{\prime}(x) = 2x - 3. \)

Key Concepts

Definition of Derivative

The derivative is a fundamental concept in calculus, which represents the rate of change of a function with respect to a variable. It's like figuring out how fast something is changing at a specific moment. To find the derivative of a function \( f(x) \) at a particular point \( x \), we use a specific mathematical formula. This formula is expressed as: \[\]

• \( f^{\prime}(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \)

Here, \( h \) is a tiny change in \( x \), and \( f(x+h) - f(x) \) is the change in the function's value due to the change in \( x \). By calculating this ratio and letting \( h \) get closer and closer to zero (infinitesimally small), we obtain the derivative. This definition is crucial as it forms the basis for more complex operations in calculus. It may seem a bit abstract initially, but once you apply it step-by-step, it becomes clearer.

Limit Process

The limit process is a method to find what value a function approaches as the variable approaches a certain point. In the context of derivatives, it helps us find the derivative by understanding how a function behaves as its input gets infinitely close to a specific value.When we have an expression like \( \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \), the limit tells us what \( \frac{f(x+h) - f(x)}{h} \) equals as \( h \) becomes extremely small.

• The expression \( \frac{f(x+h) - f(x)}{h} \) is called the difference quotient.
• As \( h \to 0 \), the difference quotient approaches the slope of the tangent to the curve at \( x \).

The limit process is important, as it moves us from a simple average rate of change to an instantaneous rate of change. By completing the limit process, we find precise and instantaneous derivatives.

Polynomial Function Differentiation

Differentiation of polynomial functions, like \( f(x) = x^2 - 3x + 5 \), involves applying the definition of the derivative. However, once we get the hang of it, it becomes intuitive and slightly mechanical.Polynomials are made up of terms like constants, \( x^n \), linear terms like \( 3x \), etc. Differentiating them involves using the power rule, which states:

• If \( f(x) = x^n \), then \( f^{\prime}(x) = n \cdot x^{n-1} \).

Here is how it applies to \( f(x) = x^2 - 3x + 5 \):

• The derivative of \( x^2 \) is \( 2x \) (since \( 2 \cdot x^{1} = 2x \)).
• The derivative of \( -3x \) is \( -3 \) (as linear terms like \( ax \) differentiate to \( a \)).
• The derivative of a constant like 5 is \( 0 \).

Thus, by using these rules, we quickly compute \( f^{\prime}(x) \) for polynomials, resulting in \( 2x - 3 \) for this specific function. Polynomial differentiation tends to be simpler due to its predictable structure.